Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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2answers
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Why is this equality involving factorials true?

$$ (n +1)! -1 +(n +1)(n +1)! = (n +2)! -1 $$ Can someone explain me how in the world is this true? :D Thanks (yes I'm trying to understand induction).
5
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1answer
99 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
2
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4answers
95 views

How many zeros are there in the number $50!$? [duplicate]

How many zeros are there in the number $50!$? My attempt: The zeros in every number come from the 10s that make up the number. The 10s are, in turn, made up of 2s and 5s. So: $\frac{50}{5*2} = 5$ ...
0
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0answers
24 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
6
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2answers
137 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
0
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0answers
36 views

Modular arithmetic, specifically n! mod m

Is there a theorem that makes solving $$ n! \equiv x \mod m $$ knowing that both $n$ and $m$ are prime? And if not, what would be the best way to go about finding $x$? cheers
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3answers
82 views

Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
1
vote
3answers
69 views

Calculate $\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$

Could I please have a hint for finding the following limit?$$\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$$
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5answers
102 views

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
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2answers
41 views

Trying to determine the number of possible combinations for a password

OVERVIEW: Making a secure password. People tend to use dictionary words as a basis for their passwords. People tend to make minor substitutions on their passwords (password -> p@$$w0rd) Assuming ...
2
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5answers
92 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...
2
votes
2answers
120 views

What is the triple factorial of a negative number, e.g., $-2$?

The triple factorial of a positive integer is computed as $7!!! = 7\cdot 4\cdot 1$. I'm interested in the value of $$(-2)!!!$$ I tried to find this value by using the Wolfram, but I found the ...
-3
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1answer
54 views

Sum of $1+(3/n!) - (9/2!) + (27/3!) - (81/4!) + (243/5!) -\ldots$ [closed]

I don't quite understand how to take care of the factorial part
4
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4answers
143 views

Simplify the expression (combination and factorial)

Simplify the following expression: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!}$ My attempt: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n+1)!}{3!(n+1-3)!} * \frac{(n-1)! + ...
0
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1answer
46 views

number of ways to choose pairs of nonadjacent people from $2k$ people sitting in a circle

The following is problem 19 in Chapter 2 from Richard Stanley's Enumerative Combinatorics, vol. 1 (2nd ed.): Suppose that $2k$ persons are sitting in a circle. In how many ways can they form pairs if ...
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0answers
13 views

Generalized superfactorial notation.

I would like to know if there's a general shorthand notation for denoting the following product: $$\mathcal{P}=a!\times (a-k)!\times (a-2k)!\times\cdots\times (a-nk)!$$ where $a$ and $k$ both ...
1
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1answer
29 views

function of input size N, combination problem [duplicate]

Can someone please elaborate how from $(N+1)+N+(N-1)+(N-2)$ one can get $= 1/2(N+1)(N+2)$? also how to prove that: $(N-1)+(N-2)+...+3+2+1+0 = \frac{N(N-1)}{2} = {N \choose 2}$ ? Thank you!
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0answers
36 views

Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
4
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2answers
184 views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
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2answers
71 views

what is remainder when $(((3!)^{5!})^{7!})^{9!…}$ is divided by 11

$$(((3!)^{5!})^{7!})^{9!...}$$ when divided by 11 what will be the reminder? Hint is appreciated Sorry I do not know how to start this problem, so I have not shown my efforts!
0
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0answers
44 views

How to Find Number of Combinations

Here is the problem: In an experiment with eight trials involving the births of three children, what is the theoretical probability that you will get the distribution of: 0 girls-once 1 girl-three ...
3
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1answer
25 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
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2answers
101 views

$2\times 5 \times 8 \ldots \times (3n-1)=?$

Does anybody know if there is a closed form expression using factorials for the above product? I'm not seeing it but I feel like there must be. The recursive relationship corresponding to this ...
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2answers
45 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
1
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1answer
26 views

Intuitive explanation for this Gamma function identity

Wolfram Alpha says that this result is true: $$\frac{\Gamma(n+1)}{\Gamma(\frac{n}{2}+1)} = \frac{\Gamma(\frac{n}{2} +\frac{1}{2})}{\Gamma(\frac{1}{2})} \times 2^n$$ This implies a curious result for ...
0
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1answer
17 views

Combining an outcome of a score

Hey I was wondering how many are the possibilities of combining the scored points of a result such $133:75$ from a basketball game? Considering that there are fouls($1$ point), normal($2$ points) and ...
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votes
1answer
40 views

What's the value of n! -1 ?? [closed]

I was practicing mathematics for my studies Then I encountered this problem Evaluate n! - 1 Edit: The problem in the book said exactly Prove: n! -1 = 1!1 + 2!2 + 3!3 ...... n!n This is all ...
3
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6answers
64 views

Prove that $2^n(n!)^2 \leq (2n)!$

Prove that $2^n(n!)^2 \leq (2n)!$ One can also use the following result to prove the above: $2 · 6 · 10 · 14 · · · · · (4n − 2) = \frac{(2n)!}{ n!}$. The above relation gives, $(2n)!=2^n n! ...
11
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3answers
99 views

Evaluating the factorial-related limit $\lim_{x \to \infty} (x + 1)!^{1 / (x + 1)} - x!^{1/x}$

I'm looking for the limit $$\lim_{x \to \infty} \left[[(x+1)!]^\frac{1}{1+x} - (x!)^\frac{1}{x}\right].$$ I've put the above in a computer program, and evaluated it at very high values of $x$ ...
1
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1answer
26 views

Calculate position in password search

I'm running a password cracker on my own password and I'm trying to calculate how long it will take. I know the rate the software is checking at and I also know the password. The password is $14$ ...
0
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2answers
47 views

Factorial Proof Problem

Suppose $m$ and $n$ are positive integers Prove $m!n! \lt (m+n)!$ I have something along the lines of: Since $1 \lt m+1$ and $2 \lt m+2$ etc.. then: $$n \lt m + n$$ So: $$n! \lt (m+n)!$$ I'm ...
2
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2answers
27 views

Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
1
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2answers
92 views

How many zeroes are there at the end of $36!^{36!}$?

Could you please tell me how many zeroes are there at the end of $36!$ to the power $36!$, i.e., $36!^{36!}$? I have been trying to find out. Read some reviews and answers related this but didn't ...
6
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2answers
473 views

Is this a meaningful approach to primes?

Motivation: I tend to be good at recognising patterns and I saw one with factorial: $$ n! = \prod_{i=1} p_i^{J(n,p_i)} $$ where $p_i$ is the $i$'th prime and $$ J(n,i)= \sum_{S=1}^\infty [n/i^S] $$ ...
1
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2answers
64 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
5
votes
4answers
94 views

Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
0
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1answer
28 views

Induction Mathematics and Factorials

\usepackage{amsmath} Evaluate the sum $\sum_{k=1}^{n} {k\over (k+1)!}$ $\sum_{k=1}^{1} {1\over (1+1)!} = {1\over 2}$ $\sum_{k=1}^{2} {2\over (2+1)!} = {5\over 6}$ $\sum_{k=1}^{3} {3\over (3+1)!} ...
4
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2answers
64 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
3
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1answer
30 views

Evaulate a determinant involving factorials.

In a problem set given by a teacher, there is the following problem. If $a_n = \frac{1}{n!}$, evaluate $$ D_n = \begin{vmatrix} a_1 & a_0 & 0 & 0 & \cdots & 0 & 0 & ...
1
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3answers
89 views

Why is Wolfram Alpha miscomputing this problem? [closed]

I was incorporating Wolfram Alpha into an API I am build, and to test it entered a few equations. One of the equations I entered was as follows. !6/(!3*!3) This ...
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2answers
27 views

A question on proving of factorial [closed]

Prove that $(n!)!$ is divisible by $(n!)^{(n-1)!}$.
6
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3answers
403 views

How do you find the factorial of a decimal or negative number and what does it show us?

I know that you can find the factorial of positive integers where n!= n(n-1)...2 x 1. However, what if you want to find the factorial of a negative integer or a decimal? I tried to do it on my ...
6
votes
1answer
62 views

Finding all the zeroes in $100!$

Is there a way to find all the $0$s in $100!$? (Including zeroes that come between two non-zero numbers) I know that to find the $0$s at the end we can use the greatest integer method. I was just ...
10
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1answer
105 views

Last nonzero digit of $2010!$ [closed]

I have to calculate the last nonzero digit of $2010!$ Till now I couldn't find any pattern.
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3answers
42 views

Stirling Formula

Find the value of $\lambda$ for this question: $\dbinom{8n}{4n} \sim \lambda \dfrac{2^{8n}}{\sqrt{n}}$ as $n \to \infty$ I tried using Stirling. Any help appreciated.
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1answer
44 views

Squeeze Theorem for Factorials

I have been having trouble with questions with factorials in Squeeze Theorem. This is the questions that I am struggling with: $\lim_{x\to \infty} {x^x\over(2x)!}$ What I have done so far: Lower ...
0
votes
1answer
69 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
1
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1answer
43 views

Relationship between Factorial and Binomial coefficients

Over at this link, there is a claim that $(2n)! = n!n! {{2n} \choose {n}}$ - see Tom Boardman's answer, the second one down. I'm wondering why this is the case and if anyone can provide a proof. Is ...
0
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2answers
73 views

Need to show following equality

I want to show that the following equality holds for any integer i,m, and n.I could not figure out how to show it analytically. Could you please help me? $$ \sum _{j=0}^n ...
1
vote
3answers
63 views

Prove that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$

I have been tasked with proving that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$ for every integer $n \geq 1$. My instinct is to use induction, but I have gotten stuck. Base Case - $n=1$: ...